There are N number of residents living and working in the economy. Everyone has the equal income I > 0
and shares the same utility function defined over private consumption C and housing unit H as follows:
$$U(C, H) = a \ln C + (1-a) H + \tilde{G},$$
where $\tilde{G} > 0$ is an exogenous variable and $a \in (0,1)$ is a parameter. The per-unit price of consumption is
denoted by $p > 0$ and per-unit price for housing is denoted by $r > 0$.
1. Set up a utility maximization problem and solve for the optimal bundle of private consumption and
housing unit.
2. Derive expressions for the own-price elasticity and cross-price elasticity of demand for housing unit
(i.e., $\epsilon_{H,r}, \epsilon_{H,p}$).
3. What is an economic interpretation of parameter $1 - a$ in this setup? Provide a reasonable number for
this parameter and justify.
4. Now, let's assume that there is a income tax of $\tau \in (0, 1)$. This means, the after-tax income for each
resident is $(1-\tau)I$. The tax revenue is used to finance government projects (e.g., water quality control,
environmental regulations, provision of parks, policing, roads, etc.). What is the total revenue? (denote
this variable as G)
